On 4/1/2013 9:03 AM, Dan wrote:> On Apr 1, 3:40 pm, Frederick Williams <freddywilli...@btinternet.com>> wrote:>> Dan wrote:>>> Real mathematicians do their own thing ... no physicist thought>>> Hilbert spaces or Riemannian geometry would have any real application>>> when they first appeared .>>>> That's a big claim to make. It seems likely that when they (Hilbert>> spaces and Riemannian geometry) first appeared, not every physicist>> voiced an opinion that has come down to us.>>>> If it was von Neumann[1] who invented Hilbert space, then it seems it>> was invented in order to give quantum mechanics a rigorous underpinning.>>>> [1] von Neumman, _Mathematical foundations of quantum mechanics_,>> Princeton UP.>>>> -->> When a true genius appears in the world, you may know him by>> this sign, that the dunces are all in confederacy against him.>> Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting>> https://en.wikipedia.org/wiki/Hilbert_space#History the concept of> Hilbert Space was developed prior to the realization of its utility> within quantum mechanics, although Von Neumann was the first to give a> completed axiomatic formulation, specifically for this purpose . My> point was that ,while I do believe set theory to be excessive, this is> not so for anything up to second order arithmetic . Furthermore , as> mathematicians , we should not let ourselves be constraint by the> narrow vision of what empiricists believe as legitimate. Leibniz ,> Euler , and Russell used infinitesimals in developing their results .> The same empiricist stigma was once manifest against the 'fictions> quantities' we now refer to as imaginary numbers . Imagine doing> modern physics without imaginary numbers. While 'empirical exploration> of numbers' may sometimes give us hints (and sometimes false ones> http://en.wikipedia.org/wiki/Graham%27s_number , as the> counterexamples are too far of to be determined empirically ) ,> mathematics isn't about empiricism, it's about rational proof . If we> proved Fermat's theorem true , we need not check every number for> counterexamples . Furthermore , doing so would be a futile endeavor .> No one has ever "seen the numbers" , or "performed an experiment on> the numbers" , unless it was fundamentally a 'thought experiment' .The> essential difference between 'thought experiment' and 'empirical> experiment' should be the theme of this discussion .

One can find a critic for everything:

"A thought experiment is no substitute for a real experiment, he claimed, and should be forbidden in science, including science education." (A paraphrase of Duhem)

The next sentence, quite naturally being the statement:

"However, in view of the important role of actual thought experiments in the history of physics ? from Galileo's falling bodies, to Newton's bucket, to Einstein's elevator ? it is unlikely that anyone will feel or should feel much sympathy for Duhem's strictures."